As mentioned in Section 16.5.1, the model allows testing for differentially expressed genes, hence for each gene, we test the hypotheses:

For a microarray with m genes, there are m null hypotheses to be tested, which implies that an adjustment for multiple testing should be applied. Throughout this chapter, we apply the false discovery rate (FDR) approach proposed by Benjamini and Hochberg (1995).

Moreover, in order to make inference about pj, there is a need to test whether the expression level of a gene and the bioassay read-out are correlated, specifically, whether the expression level of a gene can predict the bioassay read-out. Thus, in addition to the hypotheses in (16.4), one needs to test the hypotheses:

or equivalently

Under the null hypothesis, the joint model in (16.1) is reduced to

Consequently, the inference for the adjusted association can be based on a likelihood ratio test by comparing models in (16.1) and (16.7). Asymptotically, the likelihood ratio statistic follows a x2-distribution with one degree-of-freedom. The Benjamini and Hochberg (1995) procedure is used to adjust for false discovery rate when testing for the null hypotheses of H0- : Pj =0 for all the genes simultaneously per fingerprint feature.

Graphical Interpretation (II): Adjusted Association and Conditional Independence

As shown in Table 16.1, genes can be classified into subgroups according to the results obtained from the hypothesis testing in (16.4) and (16.5). For the first group of genes (a), the association between the gene expression and pIC50 exists regardless of the effect of a chemical substructure of the compound while the association from the second group of genes (b) is driven by the fingerprint feature. This association can also further expand our knowledge about the biological mechanisms of compounds to guide decision-making in lead selection. Ideally, results from the joint modeling of every fingerprint feature, gene, and activity data are generated. In this chapter, we only present the results of applying the joint model using a fingerprint feature that is mostly associated with the variation in compound activity.

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